Fixed Cross wrote:According to Spinozas logic here, a substance can only be conceived as infinite, the argument materializes in proving proposition VIII.

[list]"DEFINITIONS.

I. By that which is self—caused, I mean that of which the essence involves existence, or that of which the nature is only conceivable as existent.

What's the definition of existence?

Existence is an awareness of a being of everything apparent and conceivable.

Since it is realized through living, that existence is only a partial appearance of that everything conceivable, and even that is only a part of a totality of all, that becomes manifest, that totality is an unreachable absolute.

It is through the door of perception , that absolute manifests from birth to death.

It is that absolute that we as humans have to traverse into incarnation and re-incarnation. Every existence is another manifestation of that Absolute.

Seems like you're saying existence is relationship because in order to be aware, you must relate somehow. James said if something has no affect, then it doesn't exist and I think relationship is saying the same thing. Objective existence is therefore not existence as if there could exist a sole thing in absence of everything else.

Something can only exist in relation to something else and if it has no relation to anything else, then it can't be said to exist lest anything be said to exist, including this pink elephant sitting next to me that I can't detect.

Serendipper wrote:If we stick to the definition of the infinite as unbounded, then the opposite is the bounded.

I've already given you examples of bounded infinite sets.

No you haven't. What you have accomplished is demonstrating your inability to differentiate between boundaries of categories and boundaries within categories. If I say there are infinite apples, I am not placing boundaries on the number of apples, but that doesn't mean there are also infinite oranges and the fact there are not infinite oranges doesn't mean I've placed boundaries on apples. Due to some suspected cognitive impairment you're suffering from, you're having difficulty making this completely obvious differentiation and are running about patting yourself on the back for being totally blind.

The closed unit interval [0,1] is infinite and bounded.

Clearly you're suggesting there are unlimited numbers of numbers between 1 and 0, right? Right? Is anybody home, McFly?

Because that's what infinity means.... unlimited, unbounded numbers of things. The only bound that exists is the category of identity... which is defined by 1 and 0 per your axiomization.

So you've defined a category and stated that within that category there are an unbounded number of things.

I would feel sorry for you if you weren't so damn arrogant.

Why do you insist on using a definition that's demonstrably wrong? You can't fall back on,"Oh it's in the dictionary," since the dictionary is not authoritative on technical matters. You enjoy making up your own definitions, but that convinces no one other than you.

I can use whatever definition I want and it's total within my discretion to define terms in order to communicate. The important thing is I have defined my terms so people know what I'm talking about, unlike you who can't seem to muster a definition after repeated pleadings for you to do so, yet you continue to use a word that you can't define.

Fixed Cross wrote:S - If you dont know what existence is you have no business discussing infinity. First things first. But you knew that. So...

I have my definition of existence articulated at the top of the OP because "first things first and I knew that".

The question remains if you can read, recall what you've read, or have any clue how to define existence. Those are the unknowns.

I guess I'll take your refusal to engage this short argument by Spinoza on the grounds that you dont understand the term "existence" as admissal of trolling.

That was easy.

The whole proof seems like shit to me, which I'm certainly ready to shred to bits, but I first need to know how he defines existence. If I use my definition, then the proof quickly falls apart, but I don't know how he defines it, so I can't do anything until I get that information... which seems to be conveniently missing in light of the fact that he defined everything else under the sun except the most relevant bit.

Fixed Cross wrote:Which leaves us no one to argue against the existence of infinity. I think this closes the case, which had indeed been closed since Spinoza. He is great.

Are you warming up for a Dunning-Kruger interview or do you honestly believe that?

Serendipper wrote:No you haven't. What you have accomplished is demonstrating your inability to differentiate between boundaries of categories and boundaries within categories.

I certainly stipulate that I have no idea what that means. If it's something you made up, can you please define it? And if it's a standard subject in philosophy, can you supply a link?

Now when you say categories, of course I think of category theory, a modern foundational approach to large parts of mathematics that's an alternative to traditional set theory. Interestingly John Baez, a mathematical physicists and the original Internet math blogger, has applied n-categories, meaning categories of categories etc., to the study of loop quantum gravity in theoretical physics. So it's quite an interesting area, and one most amateur math fans haven't heard of yet.

I do not think this is what you mean, though. So why not just say what you're talking about? It certainly makes no sense to me in the context of the issue at hand.

I say again: In math, the closed unit interval on the real line is a bounded set that contains its upper and lower bound.

Furthermore, one could respond that yes, [0,1] is bounded as a metric space; but its cardinality is infinite, hence (by your argument) not bounded. However you are wrong even here. The cardinality of [0,1] is that of the reals, namely \(2^{\aleph_0}\). That cardinality is bounded below by \(\aleph_0\), and bounded above by \(2^{2^{\aleph_0}}\). So the unit interval is bounded in metric AND bounded in cardinality. It's bounded every way you can think of.

I hope that you, or at least fairminded readers, can see that I'm making substantive responses to your points. I'd appreciate substantive replies to mine.

Serendipper wrote:If I say there are infinite apples, I am not placing boundaries on the number of apples, but that doesn't mean there are also infinite oranges and the fact there are not infinite oranges doesn't mean I've placed boundaries on apples.

I perfectly well agree. If I have infinitely many apples I may well have only finitely many oranges.

Can you explain what that has to do with the boundedness of both the length and cardinality of the unit interval?

Serendipper wrote:Due to some suspected cognitive impairment you're suffering from, you're having difficulty making this completely obvious differentiation and are running about patting yourself on the back for being totally blind.

I can't speak to your upbringing or possible neurochemical imbalances.

I do address @Carleas and the other moderators of this site. If this type of discourse is ok then the site's not for me.

Moderators please advise.

Serendipper wrote:Clearly you're suggesting there are unlimited numbers of numbers between 1 and 0, right? Right?

Unlimited? No. There are exactly as many as there are real numbers. That's much much smaller than the number of possible subsets of the reals, which is less than the number of subsets of subsets of the reals, and so forth.

This is Cantor's theorem, which says that the powerset of any set has strictly larger cardinality than the set. So that in fact any transfinite cardinal you can name is bounded by the cardinality of its powerset.

So what do you think about Cantor's theorem? Do you disagree with it? If so why? I'm openminded. I don't care what position you hold if you can intelligently defend it. Say something intelligent.

Serendipper wrote:Because that's what infinity means.... unlimited, unbounded numbers of things. The only bound that exists is the category of identity... which is defined by 1 and 0 per your axiomization.

* I have certainly given no axiomitization of anything. The unit interval is a perfectly clear example to anyone who's takenn algebra II in high school.

* You have simply repeated your incorrect claim, that infinity means unbounded. That's clearly false. Repeating a claim doesn't constitute an argument in support of that claim. It only reveals you haven't got one.

* And the "category of identity?" Whats that mean? Something else you just made up?

Serendipper wrote:So you've defined a category and stated that within that category there are an unbounded number of things.

Well no. I have noted that a standard mathematical object, familiar to everyone who learned the basics of analytic geometry in high school, is infinite; yet is both bounded in length, and also bounded in cardinality.

Serendipper wrote:I would feel sorry for you if you weren't so damn arrogant.

It's funny that someone who simply knows what they're talking about appears arrogant to you.

Serendipper wrote:I can use whatever definition I want and it's total within my discretion to define terms in order to communicate.

Of course. But you don't define your terms. What's a category and how does it relate to the unit interval?

Serendipper wrote:The important thing is I have defined my terms so people know what I'm talking about,

If anyone here knows what Serendipper is talking about, please tell me. I'm openminded, if there's something I'm not getting, just explain it to me.

Serendipper wrote:unlike you who can't seem to muster a definition after repeated pleadings for you to do so, yet you continue to use a word that you can't define.

Perhaps you missed it a few days ago when I wrote:

wtf wrote:Mathematically, a set is infinite if it may be bijected to a proper subset of itself. That was one of your dictionary definitions if I recall. Galileo noted this in the 1600's and various non-Western mathematicians noted it in the 1200's or earlier.

As you can see I already defined mathematical infinity. This particular definition dates back to Dedekind in the 1880's. It's been the standard one ever since.

Serendipper, the math is beyond me, but my take is that you are not coming off well in this exchange. You seem to be insulting someone with a deeper knowledge of math than you. Doesn't mean he or she is right, but your explanations seem less grounded to this layman than yours. Are you sure you are not jumping past your own concerns that you are out of your depth and presenting a 'I am sure of what I am saying' front?

If so, just admit it, cut losses and see what you two can learn together.

I've noticed the tendency on your part to go ad hom or insulting in relation to me, rather than focusing on the substance of the issues, and it's actually good to watch it unfold in relation to someone else.

The probability of any finite string existing, according to convergence theory (which sets the bounds) is zero percent. When the infinite converges, the finite strings also converge, so infinitesimally small, that if convergence theory is true, they can't exist.

Serendipper wrote:No you haven't. What you have accomplished is demonstrating your inability to differentiate between boundaries of categories and boundaries within categories.

I certainly stipulate that I have no idea what that means.

And there we have it. How can I explain to you something that you cannot see? If you cannot differentiate between boundless quantity and boundless identity, then there is no possible way for me to show you your error. If you can't tell the difference between what something is and how many there are, then how can I help you?

If it's something you made up, can you please define it?

I've defined it over and over, but it's like a color that you can't see.

Now when you say categories, of course I think of category theory, a modern foundational approach to large parts of mathematics that's an alternative to traditional set theory. Interestingly John Baez, a mathematical physicists and the original Internet math blogger, has applied n-categories, meaning categories of categories etc., to the study of loop quantum gravity in theoretical physics. So it's quite an interesting area, and one most amateur math fans haven't heard of yet.

The category is the set of numbers between 1 and 0 of which there are infinitely many. There are no bounds to the number of numbers within the category of 0 to 1. A category is identity... the identification of what you're talking about.... a definition. If I say cat, I do not mean dog. The category is cat. There may be infinite cats, but there are not infinite categories. And just because there are not infinite categories does not mean there are limits to the number of cats. The number of cats is unbounded, but the category of cat is not; it's bounded by identity.

I do not think this is what you mean, though. So why not just say what you're talking about?

I did. Over and over. It's like explaining red to a blind man.

I say again: In math, the closed unit interval on the real line is a bounded set that contains its upper and lower bound.

Math is not the universe and math doesn't even represent it fully. Whether something applies within math is irrelevant. You could axiomize anything and then state that within that construct that certain truths apply, but it's irrelevant to the universe. Appeals to math are like appeals to authority or any other logical fallacy.

Furthermore, one could respond that yes, [0,1] is bounded as a metric space; but its cardinality is infinite, hence (by your argument) not bounded. However you are wrong even here. The cardinality of [0,1] is that of the reals, namely \(2^{\aleph_0}\). That cardinality is bounded below by \(\aleph_0\), and bounded above by \(2^{2^{\aleph_0}}\). So the unit interval is bounded in metric AND bounded in cardinality. It's bounded every way you can think of.

So your assertion is the unbounded is bounded. That's ridiculous. Make up your mind... is infinity bounded or is it not?

I hope that you, or at least fairminded readers, can see that I'm making substantive responses to your points.

Not that I can see. You're dodging and being dogmatic in asserting that the unbounded can be bounded and further demonstrating inability to see your error.

I'd appreciate substantive replies to mine.

You started it with your cocky tone in saying "I'm afraid I must dash your hope" instead of "How about we define infinity as _____________ instead of boundless."

You have taken the position that infinity exists and are dogmatically determined to defend that position even if it means asserting absurdities as truth, such as the bounded unbounded. Evidently this is your religion because it's not rooted in reason because absurdities are not reasonable. You are the dogmatist who I referred to in my OP who I didn't want to clutter the thread with pages and pages of dogmatic refusals to see reason which renders the thread useless.

Serendipper wrote:If I say there are infinite apples, I am not placing boundaries on the number of apples, but that doesn't mean there are also infinite oranges and the fact there are not infinite oranges doesn't mean I've placed boundaries on apples.

I perfectly well agree. If I have infinitely many apples I may well have only finitely many oranges.

Right, unbounded numbers of apples have the bound of being only apples and not oranges. Bounded by identity, but unbounded in quantity.

Can you explain what that has to do with the boundedness of both the length and cardinality of the unit interval?

Cardinality is just a fancy word for quantity within a set. Your pretentiousness is showing. In your example, the "length" is the set and the "cardinality" is the quantity within the set. So the set of all numbers between 1 and 0 is infinite. The set of all numbers between 1 and 2 is infinite. The set of all numbers between x and y is infinite where x and y is anything you want. Once you define x and y, you define a set with infinite cardinality.

Serendipper wrote:Due to some suspected cognitive impairment you're suffering from, you're having difficulty making this completely obvious differentiation and are running about patting yourself on the back for being totally blind.

I can't speak to your upbringing or possible neurochemical imbalances.

I do address @Carleas and the other moderators of this site. If this type of discourse is ok then the site's not for me.

Moderators please advise.

You're ignoring points!

Per the rules:

2.2 Arguments should be made in good faith: no trolling. If a moderator sees a poster presenting an argument and dismissing any counterpoints without engaging them, or suspects someone of presenting arguments purely for the sake of inflaming debate or annoying other posters, a warning may be issued.viewtopic.php?f=1&t=175550

You are "annoying other posters" by refusing to define infinity clearly while dogmatically objecting to my definition. And this definition is the same as mine: "Mathematically, a set is infinite if it may be bijected to a proper subset of itself" because the only way anything can biject with a subset of itself is if it has no bounds (which I said before and you ignored), so you said the same thing and still claim my definition is wrong.

Tell me exactly and clearly what infinity is... in plain english without math jargon that obfuscates the definition.

You are "annoying other posters" by having an inability to see the difference in boundless quantity and boundless identity.

You are "annoying other posters" by asserting the boundless has a bound.

You are not in good faith seeking the truth, but imposing your dogma and it's "annoying other posters".

If you don't know something, I'll gladly help you until you understand, but when you come across as flattering yourself for being right when you can't even see what I'm saying, then it's annoying.

Now I've linked to a Yale Phd in Mathematics and former professor at Stanford who asserts infinity doesn't exist and that should at least give you pause before you come storming on here "dashing my hopes" as if you're the grand poobah of math... and even if you were, it still wouldn't be justification to make authoritative statements about the universe.

Serendipper wrote:Clearly you're suggesting there are unlimited numbers of numbers between 1 and 0, right? Right?

Unlimited? No.

Oh, so, now there are not unlimited numbers of numbers between 1 and 0. Make up your mind! You're annoying! Is there a limit to the number of numbers between 1 and 0 or is there not?

Say something intelligent.

This from someone who asserts the unbounded has a bound

This from someone who asserts infinity is not unbounded and then provides definition which requires infinity to be unbounded.

Serendipper wrote:Because that's what infinity means.... unlimited, unbounded numbers of things. The only bound that exists is the category of identity... which is defined by 1 and 0 per your axiomization.

* I have certainly given no axiomitization of anything.

Axiom - a statement or proposition on which an abstractly defined structure is based. You stated 1,0 is the set and that's the axiom upon which we work.

The unit interval is a perfectly clear example to anyone who's takenn algebra II in high school.

I took algebra II and geometry at the same time against the recommendations of the counselors and aced them both, then was sent to a special school for mathematically gifted students, then completed college calculus before I graduated high school and on the state's dime. Admittedly, that was a long time ago and I've not kept up, but I'm not an idiot and these constant referrals to "anyone who has taken _____ should know _____" are annoying.

* You have simply repeated your incorrect claim, that infinity means unbounded. That's clearly false. Repeating a claim doesn't constitute an argument in support of that claim. It only reveals you haven't got one.

Then you better inform all the dictionaries on earth that they are incorrectly defining infinity lest the world get confused. I'm not appealing to the dictionary, but simply saying you could be famous for correcting them. Let me know what they say.

Wikipedia says: Infinity is a concept describing something without any bound

Oxford says: Limitless or endless in space, extent, or size

Dictionary.com says: Unbounded or unlimited; boundless; endless

Webster says: extending indefinitely : ENDLESS

Wtf says: the bounded unbounded

I say: wtf?

* And the "category of identity?" Whats that mean? Something else you just made up?

Yeah, kinda like the category of categories you made up.

Serendipper wrote:So you've defined a category and stated that within that category there are an unbounded number of things.

Well no. I have noted that a standard mathematical object, familiar to everyone who learned the basics of analytic geometry in high school, is infinite; yet is both bounded in length, and also bounded in cardinality.

Baloney! You said:

I ask you to consider as a mathematical example the closed unit interval [0,1], which is defined as the set of all real numbers between 0 and 1, inclusive.

We know that this set contains infinitely many real numbers; in fact, an uncountable infinity of them. Yet this set is bounded.

So you're asserting bounded unbounded; the limited unlimited, the finite infinite. You said it.

Serendipper wrote:I would feel sorry for you if you weren't so damn arrogant.

It's funny that someone who simply knows what they're talking about appears arrogant to you.

No one knows what you're talking about, including you.

Serendipper wrote:I can use whatever definition I want and it's total within my discretion to define terms in order to communicate.

Of course. But you don't define your terms.

Now you're lying.

What's a category and how does it relate to the unit interval?

A category is identity... what something is and a set to which it belongs. A unit interval is a category.

Serendipper wrote:The important thing is I have defined my terms so people know what I'm talking about,

If anyone here knows what Serendipper is talking about, please tell me. I'm openminded, if there's something I'm not getting, just explain it to me.

Openminded you are not. Dictionaries are wrong, accomplished mathematicians are wrong, my logic is wrong, but absurdities are right because you said so. I see why you named yourself wtf

Serendipper wrote:unlike you who can't seem to muster a definition after repeated pleadings for you to do so, yet you continue to use a word that you can't define.

Perhaps you missed it a few days ago when I wrote:

wtf wrote:Mathematically, a set is infinite if it may be bijected to a proper subset of itself. That was one of your dictionary definitions if I recall. Galileo noted this in the 1600's and various non-Western mathematicians noted it in the 1200's or earlier.

As you can see I already defined mathematical infinity. This particular definition dates back to Dedekind in the 1880's. It's been the standard one ever since.

Karpel Tunnel wrote:Serendipper, the math is beyond me, but my take is that you are not coming off well in this exchange.

So you're saying you have no idea, but your idea is __________.

You seem to be insulting someone with a deeper knowledge of math than you.

If "the math is beyond" you, then how can you tell? In order to judge an authority you have to be one.

Doesn't mean he or she is right, but your explanations seem less grounded to this layman than yours.

How about Yale educated and former Stanford professor of mathematics, NJ Wildberger? Is he over his head too? Heck, his whole thesis is "being grounded in reality" (limited by the size of the universe) rather than off in fantasy land like his many of his colleagues. How can I be less grounded when the premise of my argument is to be grounded (finite) rather than hinged in obscurity (infinite)?

Are you sure you are not jumping past your own concerns that you are out of your depth and presenting a 'I am sure of what I am saying' front?

Well I'm sure the bounded unbounded can't exist because it's a direct contradiction. Either infinity is bounded or it is not.

We can't say there are infinite apples and then claim there is a bound to the number of apples because there are not infinite oranges. Either the number of apples is without limit or it's not. We can't have limited unlimited. If you are not siding with me, then that is what you are asserting.

The way to show me I am wrong is to show me; not gang-up.

If so, just admit it, cut losses and see what you two can learn together.

I have no problem admitting I am wrong, but first you must show me a how something can be bounded and unbounded in quantity.

I've noticed the tendency on your part to go ad hom or insulting in relation to me, rather than focusing on the substance of the issues, and it's actually good to watch it unfold in relation to someone else.

That is an ad hom. The subject is infinity and the definition of and the existence of; not whether or not I ad hom.

This is why I said in the OP "Questions are encouraged, but dogma is not." If someone asserts and clings to absurdities, which are inherently unreasonable and therefore dogmatic, then they shouldn't be surprised if ad homs follow because the subject then transitions from discussing the differences between categories and quantities to someone's inability to see the difference.

Seriously, why not cut this shit out?

Why indeed.

Show me a quantity that is both unbounded and bounded and the shit will be cut out. Or admit that it's not possible and the shit will be cut out. Continue this dogmatism and the shit goes on.

What else can I say? I can't concede the bounded boundless exists until someone can show me how.

Until then, the definition put forth by every dictionary on earth stands, which is infinity is unbounded, limitless, endless.

If you disagree, then simply submit a CLEAR definition in PLAIN ENGLISH without obfuscating words reliant upon axioms of boundlessness.

Convey to me exactly what you think infinity is.

It's easy! Simply say "infinite = _______________"

Otherwise concede that you accept my definition. If one can do neither, then one's purpose here is to annoy, right?

If one can't accept my definition and one can't replace my definition and yet one keeps using the word while ignoring repeated pleadings to define the word they're using, then their only purpose is to be annoying. What else could it be?

Fixed Cross wrote:S - If you dont know what existence is you have no business discussing infinity. First things first. But you knew that. So...

I have my definition of existence articulated at the top of the OP because "first things first and I knew that".

The question remains if you can read, recall what you've read, or have any clue how to define existence. Those are the unknowns.

I guess I'll take your refusal to engage this short argument by Spinoza on the grounds that you dont understand the term "existence" as admissal of trolling.

That was easy.

The whole proof seems like shit to me, which I'm certainly ready to shred to bits, but I first need to know how he defines existence. If I use my definition, then the proof quickly falls apart, but I don't know how he defines it, so I can't do anything until I get that information... which seems to be conveniently missing in light of the fact that he defined everything else under the sun except the most relevant bit.

You're not fooling anyone but yourself. If that.

To be clear, what you are doing is pretend to be mentally challenged when that suits you so that you can ignore the logic that you are challenged with. It's not very impressive except in how much time and effort of yourself and others you are wasting with it.

You've been refuted about sixty times in this thread alone, but like I ambiguous, this only seems to embolden you. Lol.

Fixed Cross wrote:S - If you dont know what existence is you have no business discussing infinity. First things first. But you knew that. So...

I have my definition of existence articulated at the top of the OP because "first things first and I knew that".

The question remains if you can read, recall what you've read, or have any clue how to define existence. Those are the unknowns.

I guess I'll take your refusal to engage this short argument by Spinoza on the grounds that you dont understand the term "existence" as admissal of trolling.

That was easy.

The whole proof seems like shit to me, which I'm certainly ready to shred to bits, but I first need to know how he defines existence. If I use my definition, then the proof quickly falls apart, but I don't know how he defines it, so I can't do anything until I get that information... which seems to be conveniently missing in light of the fact that he defined everything else under the sun except the most relevant bit.

You're not fooling anyone but yourself. If that.

Thanks! Of course I'm not fooling anyone since I'm speaking sense. Those fooling folks are speaking nonsense that's imperceptible to the ones being fooled.

To be clear, what you are doing is pretend to be mentally challenged when that suits you so that you can ignore the logic that you are challenged with. It's not very impressive except in how much time and effort of yourself and others you are wasting with it.

Well at least I can type with decent grammar when insulting people

What happened to "feigning ignorance" that you used to accuse everyone of? Did you wear it out?

You've been refuted about sixty times in this thread alone, but like I ambiguous, this only seems to embolden you. Lol.

All you have to do is show me how I'm wrong and that will take the wind from my sails, if that's what you're after. Otherwise, claiming I've been refuted when I clearly haven't only flatters me that you're groping for any insult you can find in lieu of counterpoint debate because what's important is that I'm wrong, right?

I think shaming is good. I wish the community would shame not content but inability to carry out coherent dialogue with integrity. I mean, it does happen. A number of people have reacted to Prismatic who is a classice example of someone so sure they are right, they cannot acknowledge the slightest mistake and commit many of the sins I mentioned above. AT least three people have bluntly commented on his shortcomings here, after trying through many, many posts to have a rational dialogue with him. I think that kind of shaming is good. In fact I would like to see more shaming and less banning. Not that it has worked in Prismatic's case, nor am I optimistic with some of the people mentioned earlier in this thread.viewtopic.php?f=7&t=193363&p=2693911&hilit=shaming#p2693911

Now you're admonishing me for shaming him for "inability to carry out coherent dialogue with integrity".

You've also given me plenty of grief over Alan Watts' attacks on displays of emotions, so when I display emotions, you give me more flack. I just can't win with you man. I think you just have it out for me and I don't know why. I've always liked you.

Serendipper, your spending all your days wasting everyones time with your excuses for not reading any argument offered to you is not fooling anyone into thinking of you as some sort of scientist or scholar. You're quite evidently unconvinced of your own claims.

When Im confronted with a challenge I devour it. But thats my thing, I can't force that mentality on you.

Its very embarrassing that you use the verb "to be" in a question phrased to pretend you don't know how you should understand the term "existence".

Spinoza is the father of all the rationalism you admire so much. He remains one of the most respected logicians in our time. That you can't even bring yourself to read the first pages of his main work is a pretty much definitive indication of what you are.

Fixed Cross wrote:Serendipper, your spending all your days wasting everyones time with your excuses for not reading any argument offered to you

I'm fairly confident I am the only human who has read every word (multiple times) of this thread. I may be wrong, but if I could compel anyone to waste their time reading it, that would be a start.

is not fooling anyone into thinking of you as some sort of scientist or scholar.

I never said I was and would prefer not to be tainted by such labels.

You're quite evidently unconvinced of your own claims.

How so?

When Im confronted with a challenge I devour it.

Probably why you restrict yourself to deprecation instead of sinking your teeth into the debate.

But thats my thing, I can't force that mentality on you.

I'm grateful you cannot force it on me and I just hope what you have is not contagious

Its very embarrassing that you use the verb "to be" in a question phrased to pretend you don't know how you should understand the term "existence".

You caught me, but I'm not feigning ignorance because I do know how to define existence, but I do not know how HE defines existence, so I'm not feigning.

Spinoza is the father of all the rationalism you admire so much. He remains one of the most respected logicians in our time.

Let's see....

Rationalism =

a belief or theory that opinions and actions should be based on reason and knowledge rather than on religious belief or emotional response."scientific rationalism"

PHILOSOPHYthe theory that reason rather than experience is the foundation of certainty in knowledge.

THEOLOGYthe practice of treating reason as the ultimate authority in religion.

I see no distinction between reason and experience. I've long subscribed (coincidentally) to Goethe's world view that deduction is merely another mode of perception and perception/deduction are both experience: do you do it or does it happen to you? Either way, you endure it. Oops, I mean "experience it".

That you can't even bring yourself to read the first pages of his main work is a pretty much definitive indication of what you are.

It's been on my bucket list for years, but now that you've opened my eyes to him, I think I can cross him off.

To speak back to you in your language:

As opposed to what?

Anyway, I'm mostly through a critique of the Spinoza proof you submitted. I realize I'm wasting my time, but it's fun like a puzzle.

Serendipper wrote:The category is the set of numbers between 1 and 0 of which there are infinitely many. There are no bounds to the number of numbers within the category of 0 to 1.

The collection of subsets of any given set is called the powerset of the original set. Cantor's theorem shows that the cardinality of any set is strictly smaller than the cardinality of its powerset.

This is easy to see for a finite set. For example the cardinality of the set {1, 2, 3} is 3; and the cardinality of the set of subsets of {1,2,3} is 2^3, or 8. You can see this explicitly as follows. The subsets of {1,2,3} are: The empty set, {1}, {2}, {3}, (1,2}, {1,3}, {2,3}, and {1,2,3}. That's eight subsets altogether.

Cantor's theorem also true for any infinite set; and has a simple and beautiful proof given in the Wiki page I linked.

Therefore the cardinality of the set of real numbers between 0 and 1 is strictly less than the cardinality of the set of subsets of reals between 0 and 1. That is, the cardinality of the reals between 0 and 1 is BOUNDED BY the cardinality of its powerset.

So, do you either

a) Not understand Cantor's theorem? or

b) Deny it? And if so, why? What do you know that all the mathematicians in the world, from undergrad math majors to Fields medal winners, don't?

I did not understand the meaning of, "The probability of any finite string existing, according to convergence theory (which sets the bounds) is zero percent."

There are countably many finite-length strings from a finite or countably infinite alphabet. According to conventional probability theory, there is no uniform probability measure on a countable set. Can you say how you are computing these probabilities? Or do you mean that almost all binary strings have infinitely many 1 bits? The set of all binary strings having at most finitely many 1-bits is indeed zero. Is that what you mean?

I did not understand the meaning of, "The probability of any finite string existing, according to convergence theory (which sets the bounds) is zero percent."

There are countably many finite-length strings from a finite or countably infinite alphabet. According to conventional probability theory, there is no uniform probability measure on a countable set. Can you say how you are computing these probabilities? Or do you mean that almost all binary strings have infinitely many 1 bits? The set of all binary strings having at most finitely many 1-bits is indeed zero. Is that what you mean?

Imagine an infinite set that converges at 3.

There are an infinite number of these sets.

Now take any number from that set if the set is defined as infinitesimal, no problem!

But when you use convergence theory, the infinite set, algorithm, equals 3!

What's interesting about this is that both finite and infinite strings (just one) disappear on the continuum when the set converges.

This of it this way: I skipped a stone on a pond.

It's very finite.

The odds of one event occurring with infinite time is zero percent, because the infinite regresses all converge to al limit.

How do we turn 0.3... three times into the limit of one? Not the approach, but the limit ...?

At some point, the floating point infinitesimal one, makes all the others nothing, they didn't exist AT CONVERGENCE!!

Convergence theory always takes away the infinitesimals.

A sequence tending towards 1, does just that, tends towards 1, but when we actually converge it. The odds of those numbers tending towards one are so infinitesimal that unless they keep calculating as such, the tendency towards 1, in the convergence, the tendency towards 1 can no longer exist.

The bounds are zero and one, and an infinite number of sequences are moving towards zero and one, but they are neither zero or 1. Once they become zero or one, they can no longer exist.

What are the odds of correctly picking a number from an infinite sequence?

And you say now that "the cardinality of any set is strictly smaller than the cardinality of its powerset."

So, by substitution, you're saying that infinity is bounded by a bigger infinity and I'll need you to show how something bigger can be a bound for something smaller and then define what you mean by "infinity" such that said definition will apply exclusively to the reals between 0 and 1 (without also applying to the powerset since that one is different - ie bigger) and, because it's the same word, also exclusively to the powerset (without also applying to the reals set which is different - ie smaller) and then show how the powerset definition limits the reals definition (how does the big limit the small) and do so with intent to convey meaning, which means do not use obfuscating math jargon hiding the fact that your definition relies upon unbounded sets while pretending that it doesn't (for instance bijection with itself relies upon the set being unbounded).

Once you've defined your terms with a good faith effort to convey meaning and have illustrated how big things limit small things, then we'll move on.

(You only have 7 eternities to figure it out and then I'm calling time )

Fixed Cross wrote:According to Spinozas logic here, a substance can only be conceived as infinite, the argument materializes in proving proposition VIII.

Nothing can be conceived as infinite; it can only be speculated to be if one can't see that the object without borders is not an object and therefore absurd.

"DEFINITIONS.

Definition for existence is conveniently missing as are the definitions for the infinite and essence.

I. By that which is self—caused, I mean that of which the essence involves existence, or that of which the nature is only conceivable as existent.

Appealing to terms not defined. If existence were defined as relationship, clearly self-causation is impossible since at least 2 things (with illusory distinctness) are required to form a dipole (subject/object) and constitute existence. If existence is not defined as relationship, then I'm at a loss for a definition.

North and south constitute a magnet, but it's one magnet; not two distinct parts. The magnet exists as a function of the universe, but it's one universe; not universe + magnet as if the magnet could exist independent of the universe. Beyond that, I don't know.

II. A thing is called finite after its kind, when it can be limited by another thing of the same nature; for instance, a body is called finite because we always conceive another greater body. So, also, a thought is limited by another thought, but a body is not limited by thought, nor a thought by body.

Things are defined by boundaries and the size of things are irrelevant. De+finiteness constitutes thingness. And "size" itself is a construct of the spacetime fabric; size isn't something that intrinsically, timelessly, objectively exists. Time is also an emergent property.

III. By substance, I mean that which is in itself, and is conceived through itself: in other words, that of which a conception can be formed independently of any other conception.

This seems to mean things have borders which stands in opposition to the previous definition. Independence = borders. Nevermind the fact that borders join rather than separate which renders independent things impossible. All borders are held in common.

IV. By attribute, I mean that which the intellect perceives as constituting the essence of substance.

What is meant by essence? Substance? Substance of substance?

V. By mode, I mean the modifications[1] of substance, or that which exists in, and is conceived through, something other than itself.

Looks like a good definition of existence: "is conceived through something other than itself"

[1] "Affectiones"

VI. By God, I mean a being absolutely infinite—that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality.

What does infinite mean? Boundless? In that case, God is not a thing because he has no "skin" (borders). Further, if God has infinite attributes, then nonexistence is one of those attributes. Indeed, it's his only attribute!

Explanation—I say absolutely infinite, not infinite after its kind: for, of a thing infinite only after its kind, infinite attributes may be denied; but that which is absolutely infinite, contains in its essence whatever expresses reality, and involves no negation.

Absolute infinite is a oxymoron. Absolutes have definite boundaries. The number 1 is absolute, definite, and finite. It is not 1.00000000000000000000001 nor 0.9999999999999999999999, but only and absolutely 1.

VII. That thing is called free, which exists solely by the necessity of its own nature, and of which the action is determined by itself alone. On the other hand, that thing is necessary, or rather constrained, which is determined by something external to itself to a fixed and definite method of existence or action.

This contradicts itself saying free things are things determined by themselves, but determined by things external to them.

VIII. By eternity, I mean existence itself, in so far as it is conceived necessarily to follow solely from the definition of that which is eternal.

Eternity = existence? Why limit existence to just a function of 1 of 4 dimensions of a construct that already exists when instead we could define existence as infinite time AND space. Probably because time and space would have to exist before it could engender existence, which doesn't make sense. No, existence has nothing to do with spacial or temporal constructs (timeless and spaceless as opposed to infinite time and infinite space).

Explanation—Existence of this kind is conceived as an eternal truth, like the essence of a thing, and, therefore, cannot be explained by means of continuance or time,

This appears to mean time cannot come about by a process of time, but neither can existence.

though continuance may be conceived without a beginning or end.

No it can't. No one can conceive of things without beginnings or ends and "continuance" has no meaning nor relevance if there are no beginnings or ends. What's ubiquitous is irrelevant. Relevant things have beginnings and ends.

AXIOMS.

I. Everything which exists, exists either in itself or in something else.

He came close to defining existence as relationship here, except that I'm not sure how something can exist in relation to itself. Things exist in relation to what they are not. The north pole of a magnet exists in relation to the south pole and neither could exist independently because they are codependent.

II. That which cannot be conceived through anything else must be conceived through itself.

What does "conceived" mean? Thought of as? Conceptualized? So "That which cannot be conceptualized through anything else must be conceptualized through itself." Nope. Things are conceptualized by contrasting with what it is not.

Well, then does conceived mean "give birth to"? "That which cannot be birthed through anything else must be birthed through itself." Hmm... That makes no sense. There is a third option: That which cannot be birthed through anything else must not have been birthed.

See how language can obfuscate? I wish people would strive to say what they mean.

III. From a given definite cause an effect necessarily follows; and, on the other hand, if no definite cause be granted, it is impossible that an effect can follow.

How does cause influence effect? No one can answer that because it doesn't. There is no such thing as cause or effect, but they are the same process and not separate things.

The following is built upon that faulty foundation:

IV. The knowledge of an effect depends on and involves the knowledge of a cause.

What is knowledge?

V. Things which have nothing in common cannot be understood, the one by means of the other; the conception of one does not involve the conception of the other.

A way of saying existence is relative I think.

VI. A true idea must correspond with its ideate or object.

This seems to be an assertion of objective existence.

VII. If a thing can be conceived as non—existing, its essence does not involve existence.

Goes without saying.

PROPOSITIONS.

PROP. I. Substance is by nature prior to its modifications.

Proof.—This is clear from Deff. iii. and v.

If he is asserting the infinite, then where do modifications end or begin? It seems the assertion of a substance and essence undermines what he's attempting to prove.

PROP. II. Two substances, whose attributes are different, have nothing in common.

Proof.—Also evident from Def. iii. For each must exist in itself, and be conceived through itself; in other words, the conception of one does not imply the conception of the other.

This is an axiom: things with nothing in common have nothing in common.

PROP. III. Things which have nothing in common cannot be one the cause of the other.

Proof.—If they have nothing in common, it follows that one cannot be apprehended by means of the other (Ax. v.), and, therefore, one cannot be the cause of the other (Ax. iv.). Q.E.D.

Cause and effect doesn't exist and is an abstraction.

PROP. IV. Two or more distinct things are distinguished one from the other, either by the difference of the attributes of the substances, or by the difference of their modifications.

Proof.—Everything which exists, exists either in itself or in something else (Ax. i.),—that is (by Deff. iii. and v.), nothing is granted in addition to the understanding, except substance and its modifications. Nothing is, therefore, given besides the understanding, by which several things may be distinguished one from the other, except the substances, or, in other words (see Ax. iv.), their attributes and modifications. Q.E.D.

Distinct things are nonexistent to each other.

PROP. V. There cannot exist in the universe two or more substances having the same nature or attribute.

Proof.—If several distinct substances be granted, they must be distinguished one from the other, either by the difference of their attributes, or by the difference of their modifications (Prop. iv.). If only by the difference of their attributes, it will be granted that there cannot be more than one with an identical attribute. If by the difference of their modifications—as substance is naturally prior to its modifications (Prop. i.),—it follows that setting the modifications aside, and considering substance in itself, that is truly, (Deff. iii. and vi.), there cannot be conceived one substance different from another,—that is (by Prop. iv.), there cannot be granted several substances, but one substance only. Q.E.D.

There cannot exist in the universe two or more substances period. Things are substantive only in relation to something else which makes them one thing. The sun gives no light if there is nothing around to see it (photon can't leave until they already have a destination, according to Feynman).

PROP. VI. One substance cannot be produced by another substance.

Proof.—It is impossible that there should be in the universe two substances with an identical attribute, i.e. which have anything common to them both (Prop. ii.), and, therefore (Prop. iii.), one cannot be the cause of the other, neither can one be produced by the other. Q.E.D.

Yes, one distinct thing cannot produce another distinct thing or else they would not be distinct, but in fact one thing.

Corollary.—Hence it follows that a substance cannot be produced by anything external to itself. For in the universe nothing is granted, save substances and their modifications (as appears from Ax. i. and Deff. iii. and v.). Now (by the last Prop.) substance cannot be produced by another substance, therefore it cannot be produced by anything external to itself. Q.E.D. This is shown still more readily by the absurdity of the contradictory. For, if substance be produced by an external cause, the knowledge of it would depend on the knowledge of its cause (Ax. iv.), and (by Def. iii.) it would itself not be substance.

Sure I guess.

PROP. VII. Existence belongs to the nature of substances.

Proof.—Substance cannot be produced by anything external (Corollary, Prop vi.), it must, therefore, be its own cause—that is, its essence necessarily involves existence, or existence belongs to its nature.

We've already established the maximum number of things (substances, essences, whatever) to be one, so there is nothing produced and neither is there self-causation.

PROP. VIII. Every substance is necessarily infinite.

Proof.—There can only be one substance with an identical attribute, and existence follows from its nature (Prop. vii.); its nature, therefore, involves existence, either as finite or infinite. It does not exist as finite, for (by Def. ii.) it would then be limited by something else of the same kind, which would also necessarily exist (Prop. vii.); and there would be two substances with an identical attribute, which is absurd (Prop. v.). It therefore exists as infinite. Q.E.D. " --- Ethica

Every substance is necessarily infinite = Every substance is necessarily without boundaries which makes it not a substance because there is nothing that it's not.

Serendipper wrote:(You only have 7 eternities to figure it out and then I'm calling time )

I'm sorry you missed the last 140 years of mathematical progress in formalizing the infinite; which is available all over the Internet and certainly on every math or philosophy discussion forum many times over as long as the Internet's been in existence.

Your refusal to explain your either ignorance or rejection of math isn't my problem. What I find weird is that you won't even say which. You could say, "I hereby reject Cantorian infinity," and at least I'd know what you're talking about. But you act like you've never even heard of it. And that's hard to do if you follow mathematical philosophy online.

For the record and for anyone interested, Cantor's theorem (linked in my earlier post) gives us a never ending upward hierarchy of infinities, one greater than the next. There's the natural numbers, the powerset of the naturals, the powerset of the powerset of the naturals, and so forth. Each one has a larger cardinality than the one before.

And since no infinite cardinality is the largest; any individual cardinal is bounded by all the ones above it. These ideas were controversial in the 1880's, but today mathematicians have incorporated them into standard math. When you're a math major you learn that all of modern math is built on infinitary set theory. And in turn, all of physical science is based on infinitary math too. Even if the world isn't finite, the math used to describe the world depends on the mathematical theory of the infinite. If one rejects the modern mathematical theory of infinity, they reject all of the formalisms of modern physics too. That's a tough intellectual position to take.

I'm not rejecting math, except to the extent that it does not apply to physical reality, such as a completed infinity (ie the bounded unbounded). Your refusal define what you're talking about is your problem if you intend to challenge my definition and the definition of every dictionary on earth.

You can't say you reject my definition and then not offer a replacement and then continue to use the word. You also cannot offer a definition that relies on the very fact that you're disputing (a set bijected with a subset of itself requires that the set be unbounded).

If infinity is not the unbounded, then simply tell me what it is... in plain english with good faith effort to convey meaning.

What I find weird is that you won't even say which. You could say, "I hereby reject Cantorian infinity,"

I reject ALL infinities as nonexistent because they are just as absurd as your assertion that anything could be both bounded and unbounded, in terms of cardinality, at the same time.

How can the cardinality of the set 0 and 1 be "uncountable" and also be bounded? And not only that, but be bounded by a bigger unbounded thing? This is absurdity!

What is unbounded is without borders and cannot be said to exist. How can we have a box with no walls or a container with no confinement?

And all this strife and struggle because you so desperately want this concept of infinity to exist.

This debate is exactly the same as the atheist/theist debates I hear on youtube.

and at least I'd know what you're talking about. But you act like you've never even heard of it.

I've heard of it, but have no interest in it because it has no practical use, at least not to me. Arithmetic, algebra, geometry, trigonometry, calculus all have uses that apply to the world I live in and I've never needed set theory. I've read that there are uses for it in computer programming to counter infinite loops, but I'm not a programmer and perhaps there could be another way which doesn't involve set theory.

And that's hard to do if you follow mathematical philosophy online.

I follow math from the engineering perspective as it relates to reality. The purpose of this thread is to establish if infinity exists and not if it exists in manmade constructs. I could write a play containing pink elephants and then claim pink elephants exist and that is essentially what you're doing by appealing to math.

For the record and for anyone interested, Cantor's theorem (linked in my earlier post) gives us a never ending upward hierarchy of infinities, one greater than the next. There's the natural numbers, the powerset of the naturals, the powerset of the powerset of the naturals, and so forth. Each one has a larger cardinality than the one before.

For anyone interested, there is some controversy about this:

"I protest against the use of inﬁnite magnitude as something completed, which is never permissible in mathematics. Inﬁnity is merely a way of speaking, the true meaning being a limit which certain ratios approach indeﬁnitely close, while others are permitted to increase without restriction." Johann Carl Friedrich Gauss - a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the Princeps mathematicorum (Latin for "the foremost of mathematicians") and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians. https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

Who is more authoritative than Gauss? Only Euler and he existed too early to chime in.

"I don’t know what predominates in Cantor’s theory - philosophy or theology, but I am sure that there is no mathematics there." Leopold Kronecker - a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by Weber (1893) as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man.") https://en.wikipedia.org/wiki/Leopold_Kronecker

And since no infinite cardinality is the largest; any individual cardinal is bounded by all the ones above it.

Can't you see that makes no sense? If no infinite cardinality is the largest, then there is no bound; therefore it can't be bounded; especially by something bigger that is also unbounded. This is like saying our infinite universe is bounded by a larger infinite universe when neither have any bound.

These ideas were controversial in the 1880's, but today mathematicians have incorporated them into standard math.

The reasons for that are pretty obvious once one takes into account the religious aspects.

When you're a math major you learn that all of modern math is built on infinitary set theory.

I don't see how 2+2 depends on infinite set theory. I don't see how A=PI(r^2) depends on infinite set theory. I don't see how ax^2 + bx +c = y depends on infinite set theory. I don't even see how integral calculus depends on set theory.

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." — Albert Einstein

Even if the world isn't finite, the math used to describe the world depends on the mathematical theory of the infinite. If one rejects the modern mathematical theory of infinity, they reject all of the formalisms of modern physics too. That's a tough intellectual position to take.

I find the assertion that the unbounded exists in physical reality is a tough position to take, especially when it cannot be tested scientifically nor even conceived cognitively, much like god, and must be held on faith.

The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.

Serendipper wrote:You can't say you reject my definition and then not offer a replacement and then continue to use the word.

I already defined infinity several times. A set is infinite if it can be bijected with a proper subset of itself. That definition is due to Dedekind in the 1880s.

Serendipper wrote: You also cannot offer a definition that relies on the very fact that you're disputing (a set bijected with a subset of itself requires that the set be unbounded).

But nothing could be further from the truth. There are lots of bounded infinite sets. I gave the unit interval as one example. Another is the ordinal \(\omega + 1\). That's the usual set of natural numbers 0, 1, 2, 3, ... with an extra element at the end, like this: 0, 1, 2, 3, 4, ..., \(\omega\). That's an infinite set that contains a largest element. It's bounded.

You keep insisting that infinite sets must be unbounded, but there are so many counterexamples that you're simply wrong, absent a compelling argument. I have no problem with you holding different opinions than standard math. I do have a problem that you can't intelligently defend your position, or frankly even articulate it.

Serendipper wrote:If infinity is not the unbounded, then simply tell me what it is... in plain english with good faith effort to convey meaning.

A set is infinite if it may be placed into bijection with at least one of its proper subsets. Galileo noted this fact about the natural numbers and there are references in Indian mathematics hundreds of years earlier. https://en.wikipedia.org/wiki/Galileo%27s_paradox. This is not a new idea.

Serendipper wrote:I reject ALL infinities as nonexistent because they are just as absurd as your assertion that anything could be both bounded and unbounded, in terms of cardinality, at the same time.

I see. So earlier when you wrote ...

Serendipper wrote:The category is the set of numbers between 1 and 0 of which there are infinitely many.

... you were just kidding? You have contradicted yourself. You reject infinity and accept infinity.

Serendipper wrote:How can the cardinality of the set 0 and 1 be "uncountable" and also be bounded?

Easy. We know its cardinality is uncountable, \(2^{\aleph_0}\) in fact. And its powerset has cardinality \(2^{2^{\aleph_0}}\), which is strictly greater.

Serendipper wrote:And not only that, but be bounded by a bigger unbounded thing? This is absurdity!

You are not the first person to have felt that way. If you wish to reject all of modern mathematics from 1874 onward, that's perfectly ok. Back in those days many people rejected these ideas. Today they're almost universally accepted. But I did say "almost." There are finitists who reject the axiom of infinity. The only problem with that doctrine is that it's difficult and presently impossible to found physical science! You can't do relativity and quantum physics without infinitary math. Now that is in fact a genuine philosophical problem. I don't think you appreciate it though. You wave your hands and say, "absurd," but you haven't proposed how to salvage physics; and you are decidedly out of step with modern math. Nothing wrong with that, but you have the burden of justifying your position. Everyone else accepts infinitary math because it is (1) interesting; and (2) useful. Whether it's "true" in any meaningful sense, I cannot say.

So you see I am not disagreeing with you. I am only pointing out some of the subtle issues with your position that you may not have considered.

Serendipper wrote:What is unbounded is without borders and cannot be said to exist. How can we have a box with no walls or a container with no confinement?

I make no claim that any mathematical objects "exist." Existence is a technical term. If we can prove something exists by our axioms, it has mathematical existence. I make no further ontological claims. You're arguing with a strawman.

Do I claim the unit interval of real numbers exists in the physical universe? Of course not. Do I claim it exists in some sort of abstract Platonic sense? No not even that!

I only claim it has mathematical existence. I make the same claim for the number 3. That has mathematical existence. I have no idea if it has Platonic existence. You're fighting a strawman argument that you've made up yourself.

Serendipper wrote:And all this strife and struggle because you so desperately want this concept of infinity to exist.

No I make no claims of existence at all. Only of mathematical existence within infinitary set theory. That's a very limited claim and it is all I claim. Just as when I play chess, I don't claim the knight "really" moves that way. It only moves that way as long as I accept the rules of the game.

Mathematical infinity only works if we accept the rules of the game of set theory. But this game is (1) interesting; and (2) useful. So we play the game.

I believe this ontological point is the heart of our disagreement. I make no claims whatever for the "truth" or "existence" of mathematics. I only claim interestingness, usefulness, and the agreement of almost all modern mathematicians on these matters.

I make no claim about truth or Platonic or physical existence.

Serendipper wrote:This debate is exactly the same as the atheist/theist debates I hear on youtube.

No it's not. I have no "belief." I only enjoy playing the game. If I accept the rules of chess, I may move my knight in such-and-so a manner when it's my turn to move. If I accept the rules of set theory, I may prove various facts about infinite sets.

That's all I'm claiming. You are the one fighting against a position I don't hold.

Serendipper wrote:I've heard of it, but have no interest in it because it has no practical use, at least not to me.

Without infinitary set theory you lose quantum physics, which is based on a mathematical discipline called functional analysis. You simply lose QM. Relativity too. A century of physics, gone.

Serendipper wrote:Arithmetic, algebra, geometry, trigonometry, calculus all have uses that apply to the world I live in and I've never needed set theory.

I've read that there are uses for it in computer programming to counter infinite loops, but I'm not a programmer and perhaps there could be another way which doesn't involve set theory.

Ah. Well calculus can not be placed on a logically sound basis without infinitary set theory. That was the great advance of the 1880's, what's called in math history the arithmetization of analysis.

You don't need set theory to do freshman calculus. You do need it to do higher mathematical analysis. That was the lesson of the 1800's and 1900's.

You are confusing the terminal point of your own education with the terminal point of human knowledge.

Serendipper wrote:I follow math from the engineering perspective as it relates to reality.

I have no problem with that. But why are you on a philosophy discussion forum talking about things you don't understand?

Serendipper wrote: The purpose of this thread is to establish if infinity exists and not if it exists in manmade constructs.

From an engineering perspective? You're several disciplines removed. From engineering to applied physics to theoretical physics to mathematics to philosophy.

You can not hope to elucidate the infinite from an engineering perspective.

Serendipper wrote:I could write a play containing pink elephants and then claim pink elephants exist and that is essentially what you're doing by appealing to math.

But no. You could write a play about pink elephants and if your play was entertaining, people would enjoy it. And it would be useful in the sense that it brings joy to children. Again these two criteria: Interestingness and usefulness.

I appeal to the interestingness and usefulness of math. I make no claims about truth or existence.

Serendipper wrote:For anyone interested, there is some controversy about this:

Posting that hoary old quote from Gauss is the last refuge of someone with no argument to make. A lot has happened since the 1820's when Gauss did his work.

Likewise Kronecker. Kronecker was wrong, and Hilbert was right when Hilbert said, "No one shall expel us from Cantor's paradise." And quoting historical figures out of context really doesn't make you look clever. It makes you look like someone who can surf Wiki but hasn't got any background or knowledge of the historical developments.

Serendipper wrote:Can't you see that makes no sense? If no infinite cardinality is the largest, then there is no bound; therefore it can't be bounded; especially by something bigger that is also unbounded. This is like saying our infinite universe is bounded by a larger infinite universe when neither have any bound.

You don't even know what the word bounded means. Any infinite cardinal is bounded by that of its powerset.

Serendipper wrote:You can't say you reject my definition and then not offer a replacement and then continue to use the word.

I already defined infinity several times. A set is infinite if it can be bijected with a proper subset of itself. That definition is due to Dedekind in the 1880s.

Therefore you agree with me that the infinite is the unbounded.

Serendipper wrote: You also cannot offer a definition that relies on the very fact that you're disputing (a set bijected with a subset of itself requires that the set be unbounded).

But nothing could be further from the truth.

Nothing can be closer to the truth. The ONLY way to biject a set with a subset of itself is to have an unbounded set. You cannot do it with a bounded set. Therefore the definition of the infinite is unbounded and your definition is just another way of saying that.

There are lots of bounded infinite sets. I gave the unit interval as one example. Another is the ordinal \(\omega + 1\). That's the usual set of natural numbers 0, 1, 2, 3, ... with an extra element at the end, like this: 0, 1, 2, 3, 4, ..., \(\omega\). That's an infinite set that contains a largest element. It's bounded.

If it's bounded, it's not infinite; if it's infinite, it's not bounded. That remains true until you come up with a new definition for infinity that does not require unbounded sets.

You keep insisting that infinite sets must be unbounded,

Because they are called infinite. If infinite doesn't mean unbounded, then what does it mean?

absent a compelling argument.

I have no argument against dogma.

I have no problem with you holding different opinions than standard math. I do have a problem that you can't intelligently defend your position, or frankly even articulate it.

First you say I suck at math and now my linguistics are bad too?

Serendipper wrote:If infinity is not the unbounded, then simply tell me what it is... in plain english with good faith effort to convey meaning.

A set is infinite if it may be placed into bijection with at least one of its proper subsets. Galileo noted this fact about the natural numbers and there are references in Indian mathematics hundreds of years earlier. https://en.wikipedia.org/wiki/Galileo%27s_paradox. This is not a new idea.

I'm sure Galileo would have been smart enough to see that in order for bijection to apply, the sets would need to be unbounded.

Serendipper wrote:I reject ALL infinities as nonexistent because they are just as absurd as your assertion that anything could be both bounded and unbounded, in terms of cardinality, at the same time.

I see. So earlier when you wrote ...

Serendipper wrote:The category is the set of numbers between 1 and 0 of which there are infinitely many.

... you were just kidding? You have contradicted yourself. You reject infinity and accept infinity.

Even though I assert infinity doesn't exist, I can still talk about it in the context of an example. For instance I don't believe Yahweh exists, but I can recite many things he said in the bible.

In your example, the category is the set of numbers between 1 and 0 of which there are infinitely many.

Serendipper wrote:How can the cardinality of the set 0 and 1 be "uncountable" and also be bounded?

Easy. We know its cardinality is uncountable, \(2^{\aleph_0}\) in fact. And its powerset has cardinality \(2^{2^{\aleph_0}}\), which is strictly greater.

You didn't answer the question. If the uncountable is bounded, then what is the bound? At what point must I stop counting because I hit your bound?

Serendipper wrote:And not only that, but be bounded by a bigger unbounded thing? This is absurdity!

You are not the first person to have felt that way. If you wish to reject all of modern mathematics from 1874 onward, that's perfectly ok.

I don't have to reject all of modern mathematics just because I don't believe in absurdities.

Back in those days many people rejected these ideas. Today they're almost universally accepted. But I did say "almost." There are finitists who reject the axiom of infinity. The only problem with that doctrine is that it's difficult and presently impossible to found physical science! You can't do relativity and quantum physics without infinitary math. Now that is in fact a genuine philosophical problem. I don't think you appreciate it though.

We don't need infinity for relativity and I'm not a big fan of quantum physics.

You wave your hands and say, "absurd," but you haven't proposed how to salvage physics;

Just because we use a concept to arrive at a conclusion that matches reality does not mean that the concept exists in reality.

Nothing wrong with that, but you have the burden of justifying your position.

I'm not the one asserting the existence of something that cannot be tested.

I make no claim that any mathematical objects "exist."

You're arguing with a strawman.

I only claim it has mathematical existence.

I make no claims whatever for the "truth" or "existence" of mathematics.

I only claim interestingness, usefulness, and the agreement of almost all modern mathematicians on these matters.

I make no claim about truth or Platonic or physical existence.

You are the one fighting against a position I don't hold.

I don't really believe any of that, but I'll remember you said it for future reference.

If you had simply claimed that infinity exists only within a mathematical construct like the knight exists only in chess, then I would have agreed. The point of the thread was that infinity can't be said to exist in what we call reality.

Ah. Well calculus can not be placed on a logically sound basis without infinitary set theory. That was the great advance of the 1880's, what's called in math history

I had 6 courses of calculus in college and don't remember needing set theory. I looked in my book and found no reference to set theory in the table of contents nor the index. This is my book https://www.amazon.com/Calculus-Analyti ... 0201163209 (I can't believe it's still worth $50 lol)

It's a good textbook and has about 1200 pages. Inside on page 70 it says "While there is no real number 'infinity', the word 'infinity' provides a useful language for describing how some functions behave when their domains or ranges exceed all bounds."

Number theory sucked. I just wasn't interested for lack of having practical purpose or maybe I was burned out on math by that point... or both.

why are you on a philosophy discussion forum talking about things you don't understand?

Why would I talk about it if I understood it?

"Those who know, don't say; those who say, don't know." - Lao Tzu, Confucius or someone like that."Writing is the act of discovering what you believe" - David Hare

I appeal to the interestingness and usefulness of math. I make no claims about truth or existence.

So you go from that ^

To this:

Posting that hoary old quote from Gauss is the last refuge of someone with no argument to make. A lot has happened since the 1820's when Gauss did his work.

Likewise Kronecker. Kronecker was wrong, and Hilbert was right when Hilbert said, "No one shall expel us from Cantor's paradise." And quoting historical figures out of context really doesn't make you look clever. It makes you look like someone who can surf Wiki but hasn't got any background or knowledge of the historical developments.

Looks like you're making claims to me.

I don't need to rely on Gauss to substantiate my argument, but as an aside I was showing there are those who disagree with the concept of infinity, including the princeps mathematicorum.

Serendipper wrote:Can't you see that makes no sense? If no infinite cardinality is the largest, then there is no bound; therefore it can't be bounded; especially by something bigger that is also unbounded. This is like saying our infinite universe is bounded by a larger infinite universe when neither have any bound.

You don't even know what the word bounded means.

Then define it.

Any infinite cardinal is bounded by that of its powerset.

How does a big thing limit a smaller thing?

Serendipper wrote:NJ Wildberger ...

LOLOLOLOL. I stopped reading here.

Arrogance and ignorance go hand in hand.

Wildberger's ideas on infinity are universally regarded as cranky.

Which means nothing. More appeals to popularity on your part. Are we teenage girls seeking to be fashionable?

All the best. I get that you like to surf Wiki.

Wildberger was on the OP. If you were going to stop upon seeing Wildberger, then why did you begin?

You would be better off learning some math or philosophy.

You would be better off updating your insults. At least make them funny.

wtf wrote:I already defined infinity several times. A set is infinite if it can be bijected with a proper subset of itself. That definition is due to Dedekind in the 1880s.

Therefore you agree with me that the infinite is the unbounded.

I didn't read any further in your post. I'll just leave this here for people to read and let them draw their own conclusions.

You know it occurs to me that perhaps you don't know what the word unbounded means. An ordered set \(S\) is bounded (above, say) if there is some element \(x\), which may or may not be in \(S\), such that if \(s \in S\) then \(s \leq x\).

By that definition the closed unit interval \([0,1]\) is bounded above by 1, which is in that set; or by 2, which isn't. Either way it's bounded. Likewise each cardinal up to that of the real numbers, \(2^{\aleph_0}\), is less than the cardinal of the powerset of the reals, \(2^{2^{\aleph_0}}\). And the third example I've given is the ordinal \(\omega\), an upper bound for the set of natural numbers. It's an ordinal number greater than each of 0, 1, 2, 3, 4, ... The first example should be obvious to anyone who made it out of high school analytic geometry. The latter two examples are less familiar but not very difficult.

I'm starting to think that you mean something entirely different by the word bounded. Because a set may be bijected with a proper subset of itself yet still be bounded by my definition. I just gave three examples. The definition I gave is the standard one in math.

In math, the attributes of bounded and infinite are not equivalent. In fact if a set is finite, it's bounded. So you do have an implication in one direction. Finite implies bounded. But the other direction fails. A bounded set MAY be finite; or it may be infinite.

To sum up what I said earlier, you don't need to accept modern math. But you do have to give it its due. If you have some different definition for the word bounded, by all means provide it.

You know it occurs to me that perhaps you don't know what the word unbounded means.

Unbounded means having no bounds or limits; unlimited; infinite.

An ordered set \(S\) is bounded (above, say) if there is some element \(x\), which may or may not be in \(S\), such that if \(s \in S\) then \(s \leq x\).

X is an arbitrary starting point in an infinite series that has neither intrinsic beginning nor end.

By that definition the closed unit interval \([0,1]\) is bounded above by 1, which is in that set; or by 2, which isn't. Either way it's bounded.

Then it's not infinite. The infinite cannot have a beginning nor an end.

Likewise each cardinal up to that of the real numbers, \(2^{\aleph_0}\), is less than the cardinal of the powerset of the reals, \(2^{2^{\aleph_0}}\).

Insisting there is a cardinality greater than that of all natural numbers is placing a limit on natural numbers that is transcended by the greater cardinality and if there is a limit to the natural numbers that is transcended by another cardinality, it would be a cinch for you to display it right here: what exactly is the biggest natural number that is transcended?

You're struggling in transparent desperation to show that the absurd is true, that the unlimited has a limit that can be transcended by a bigger unlimited thing.